Extrude a cylinder from the side of another cylinder
Maybe the easiest way is to use the Knife Project tool:


Duplicate the top of your cylinder, rotate the circle and make it another object.
Go in front view, select the circle, then the cylinder, and go in Edit mode.
Press the Knife Project button on the Tools panel (on your left, activated with T).
Now you have a circle marked on your cylinder. You can get rid of you circle object.
Delete the faces of this circle.
Extrude the edges.
Flatten on the Y axis with a SY0.
Fill the hole with F.
As it's better to only have quads, give an edge loop around the cylinder, use the knife: K to activate, Z to cut through, C to cut perpendicularly.
You have what you need (I guess).